We study the transient dynamics of single species reaction diffusion systemswhose reaction terms $f(u)$ vary nonlinearly near $upprox 0$, specifically as$f(u)pprox u^{2}$ and $f(u)pprox u^{3}$. We consider three cases, calculatetheir traveling wave fronts and speeds emph{analytically} and solve theequations numerically with different initial conditions to study the approachto the asymptotic front shape and speed. Observed time evolution is found to bequite sensitive to initial conditions and to display in some cases nonmonotonicbehavior. Our analysis is centered on cases with $f'(0)=0$, and uncoversfindings qualitatively as well quantitatively different from the more familiarreaction diffusion equations with $f'(0)>0$. These differences are ascribableto the disparity in time scales between the evolution of the front interior andthe front tail.
展开▼
机译：我们研究了单种反应扩散系统的瞬态动力学，该系统的反应项$ f（u）$在$ u approx 0 $附近非线性变化，特别是$ f（u） approx u ^ {2} $和$ f（u） approx u ^ {3} $。我们考虑了三种情况，以解析的方式计算它们的行波波前和速度，并用不同的初始条件数值求解方程，以研究渐近波前形状和速度的方法。发现观察到的时间演变对初始条件相当敏感，并且在某些情况下显示为非单调行为。我们的分析集中于$ f'（0）= 0 $的情况，并且定性地发现了定量上与$ f'（0）> 0 $的反应扩散方程不同的发现。这些差异可归因于前内饰和前尾翼演变之间的时标差异。
展开▼
